Characterization of Pseudo-Boolean Models by Boolean Models and Its Applications to Intermediate Logics

Abstract

For the study of intermediate logics, pseudo-Boolean algebras play a very important role as their models. So an investigation into the algebraic structure of pseudo-Boolean models seems essential. For dealing with these models, we already know two operations on models, i.e., Cartesian product and the pile operation. But these operations are incomplete in the sense that there exist finite models which can not be obtained from the two element model Sl by these operations alone. There has been a problem of finding a complete set of operations in this sense. (See Hosoi QQ, and Hosoi and Ono Q8].) Our main result (Theorem 3.7) solves this problem. More precisely, in §2, we shall introduce the notion of the patch operation on models, and in §3, we shall show that Cartesian product and the patch operation are complete in the sense that any finite model can be obtained from S1 by these operations. Further, we shall study intermediate logics through pseudo-Boolean models. The notion of slice defined axiomatically by Hosoi will be characterized algebraically in §4. To do this, we shall define the notion of the height of pseudo-Boolean models. We shall prove that this height corresponds to the index of slice to which belongs the logic characterized by the model. In §5, we shall apply the main result to obtain an easy method for counting the height of models, and a theorem on the immediate predecessors of certain logics.